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Numerical Expression Evaluation

Use properties of equality and order of operations.

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Numerical Expression Evaluation
License: CC BY-NC 3.0

Boris is spending the day doing stock inventory at his dad’s fishing and bait store. Boris writes the following expression to represent the fresh bait in the window that day:

\begin{align*}27+2 \times 5-3+8+7=49\end{align*}27+2×53+8+7=49

Boris’s dad tells him that all the numbers are correct, except that the total number of packages of fresh bait is only 33, not 49. If Boris has all the right numbers, that must mean he has made a mistake in his order of operations. Where can Boris insert parentheses to get the right answer and make this a true expression?

In this concept, you will learn how to use order of operations to identify and create numerical expressions that are true.

Evaluating Numerical Expressions

Sometimes you will need to use the order of operations to determine whether or not a numerical expression is true. Sometimes inserting a grouping symbol, like parentheses, in the correct spot can help you to make an expression true. 

Let's review the order of operations.

 Order of Operations

P - parentheses

E - exponents

MD - multiplication or division, in order from left to right

AS - addition or subtraction, in order from left to right

Now let's consider an expression that is missing parentheses. You will need to insert parentheses into this expression in order to make it true.

\begin{align*}4 + 2 \times 3 + 7 - 5 + 8^2 = 76\end{align*}4+2×3+75+82=76

Since exponents and multiplication and division come before addition and subtraction, it makes sense that you will need to use parentheses to group some part of addition or subtraction so that this operation is completed first.

Let's try inserting parentheses around \begin{align*}7-5\end{align*}75.

\begin{align*}4 + 2 \times 3 + (7 - 5) + 8^2 = 76\end{align*}4+2×3+(75)+82=76


First, complete the operation in parentheses.

\begin{align*}4 + 2 \times 3 + (7 - 5) + 8^2 = & 76 \quad \text{Evaluate }(7-5)=2\\ 4 + 2 \times 3 + 2 + 8^2 = & 76 \\\end{align*}4+2×3+(75)+82=4+2×3+2+82=76Evaluate (75)=276Then, evaluate the exponent.

\begin{align*}4 + 2 \times 3 + 2 + 8^2 = & 76 \quad \text{Evaluate }8^2=64\\ 4 + 2 \times 3 + 2 + 64 = & 76 \\\end{align*}4+2×3+2+82=4+2×3+2+64=76Evaluate 82=6476

Next, multiply.

\begin{align*}4 + 2 \times 3 + 2 + 64 = & 76 \quad \text{Multiply }2 \times 3 = 6\\ 4 + 6 + 2 + 64 = & 76 \\\end{align*}4+2×3+2+64=4+6+2+64=76Multiply 2×3=676

Then, add and subtract in order from left to right.

\begin{align*}4 + 6 + 2 + 64 = & 76 \quad \text{Add }4+6=10\\ 10+2+64 = & 76 \quad \text{Add }10+2=12\\ 12+64 = & 76 \quad \text{Add }12+64 = 76\\ 76 = & 76 \quad \text{This is true}\\\end{align*}4+6+2+64=10+2+64=12+64=76=76Add 4+6=1076Add 10+2=1276Add 12+64=7676This is true

The simplified equation is 76 = 76, which is true, so inserting parentheses around 7 - 5 makes this a true expression.

Look at another expression that is missing parentheses.

\begin{align*}5 + 3 \times 2 + 7 - 1 = 22\end{align*}5+3×2+71=22

How do you know it's missing parentheses? Because if you evaluate this problem as it is, following order of operations, you get an untrue equation. Here is how that would look:

\begin{align*}\color{red}5 + 3 \times 2 + 7 - 1 = & \color{red}22\\ \color{red}5+6+7-1 = & \color{red}22\\ \color{red}11+7-1 = & \color{red}22\\ \color{red}18 - 1 = & \color{red}22 \quad \text{This is an untrue equation}\\\end{align*}5+3×2+71=5+6+71=11+71=181=22222222This is an untrue equation

Evaluating the operations in a different order will change the outcome, so try putting parentheses around 2 + 7.

\begin{align*}\color{red}5 + 3 \times (2 + 7) - 1 = 22\end{align*}5+3×(2+7)1=22

First, complete the addition in parentheses.

\begin{align*}\color{red}5 + 3 \times (2 + 7) - 1 = & \color{red}22 \quad \text{Evaluate }(2+7)=9\\ \color{red}5+3 \times 9 - 1 = & \color{red}22\\\end{align*}5+3×(2+7)1=5+3×91=22Evaluate (2+7)=922 

Then, as there are no exponents, complete the multiplication.

\begin{align*}\color{red}5+3 \times 9 - 1 = & \color{red}22 \quad \text{Multiply }3\times 9=27\\ \color{red}5+27-1 = & \color{red}22\\\end{align*}5+3×91=5+271=22Multiply 3×9=2722

Next, complete the addition and subtraction in order from left to right.

\begin{align*}\color{red}5+27-1= & \color{red}22 \quad \text{Add }5+27=32\\ \color{red}32-1= & \color{red}22 \quad \text{Subtract }32-1=31\\ \color{red}31 = & \color{red}22 \quad \text{This is an untrue statement}\\\end{align*}5+271=321=31=22Add 5+27=3222Subtract 321=3122This is an untrue statement

The simplified equation 31 = 22 is clearly incorrect. Inserting parentheses around \begin{align*}2+7\end{align*}2+7 does not make this a true expression.

Try putting the parentheses around 5 + 3.

\begin{align*}(5 + 3) \times 2 + 7 - 1 = 22\end{align*}(5+3)×2+71=22

First, complete the addition in parentheses. 

\begin{align*}(5 + 3) \times 2 + 7 - 1 = & 22 \quad \text{Evaluate }(5+3) = 8\\ 8 \times 2+7-1 = & 22\\\end{align*}(5+3)×2+71=8×2+71=22Evaluate (5+3)=822

Then, there are no exponents, so complete the multiplication.

\begin{align*}8 \times 2 + 7 - 1 = & 22 \quad \text{Multiply }8 \times 2 = 16\\ 16+7-1 = & 22\\\end{align*}8×2+71=16+71=22Multiply 8×2=1622

Finally, complete the addition and subtraction in order from left to right.

\begin{align*}16+7-1=&22 \quad \text{Add }16+7=23\\ 23-1=&22 \quad \text{Subtract }23-1=22\\ 22 = & 22 \quad \text{This is true}\\\end{align*}16+71=231=22=22Add 16+7=2322Subtract 231=2222This is true

The simplified equation is 22 = 22. Inserting parentheses around 5 + 3 makes this a true statement.

Examples

Example 1

Earlier, you were given a problem about Boris and his bait expression.

Boris needs to add parentheses somewhere to make a true expression out of:

\begin{align*}27+2 \times 5 - 3 + 8 +7 = 33\end{align*}27+2×53+8+7=33

First, try inserting parentheses around 27+2.

\begin{align*}(27+2) \times 5-3+8+7 = & 33\\ 29 \times 5 - 3 + 8 + 7 = & 33\\ 145 - 3 + 8 + 7 = & 33\\ 142+8+7 = & 33\\ \color{red}157 = & \color{red}33 \quad \text{This is untrue}\\\end{align*}(27+2)×53+8+7=29×53+8+7=1453+8+7=142+8+7=157=3333333333This is untrue

Since 157 is not equal to 33, putting parentheses around 27 + 2 did not work.

Next, try inserting parentheses around 3 + 8.

\begin{align*}27+2 \times 5 - (3 + 8) + 7 = & 33\\ 27 + 2 \times 5 - 11+7 = & 33\\ 27 + 10 - 11 + 7 = & 33\\ 37 - 11 + 7 = & 33\\ 26 + 7 = & 33\\ 33 = & 33 \quad \text{This is true.}\\\end{align*}27+2×5(3+8)+7=27+2×511+7=27+1011+7=3711+7=26+7=33=333333333333This is true.

The answer is \begin{align*}27+2 \times 5-(3+8)+7=33\end{align*}27+2×5(3+8)+7=33.

Boris now has a true expression for the total number of packages of fresh bait in the window.

Example 2

Determine whether this is a true expression.

\begin{align*}7\times \left(4+1\right)-7\times 2=21\end{align*}7×(4+1)7×2=21

First, evaluate the parentheses.

\begin{align*}7 \times \left( 4+1 \right) -7\times 2=&21 \quad \text{Evaluate }(4+1)=5\\ 7 \times 5 - 7 \times 2 = & 21\\\end{align*}7×(4+1)7×2=7×57×2=21Evaluate (4+1)=521

Next, there are no exponents to evaluate, so multiply.

\begin{align*}7 \times 5 - 7 \times 2 = & 21 \quad \text{Multiply }7 \times 5 = 35\\ 35 - 7 \times 2 = & 21 \quad \text{Multiply }7 \times 2 = 14\\ 35 - 14 = & 21\\\end{align*}7×57×2=357×2=3514=21Multiply 7×5=3521Multiply 7×2=1421

Finally, subtract.

\begin{align*}35 - 14 = & 21 \quad \text{Subtract }35 - 14 = 21\\ 21 = & 21 \quad \text{This is true}\\\end{align*}3514=21=21Subtract 3514=2121This is true

The answer is "The expression is true."

Example 3

\begin{align*}{3}^{2}+{4}^{2}-9+\left(3\times 2\right)=22\end{align*}32+429+(3×2)=22 First, evaluate the parentheses.

\begin{align*}{3}^{2}+{4}^{2}-9+\left(3\times 2\right)= & 22 \quad \text{Evaluate }(3 \times 2)=6\\ 3^2+4^2-9+6= & 22\\\end{align*}32+429+(3×2)=32+429+6=22Evaluate (3×2)=622 Next, consider the exponents.

\begin{align*}{3}^{2}+{4}^{2}-9+6= & 22 \quad \text{Evaluate }3^2=9\\ 9+{4}^{2}-9+6= & 22 \quad \text{Evaluate }4^2=16\\ 9+16-9+6 = & 22\\\end{align*}32+429+6=9+429+6=9+169+6=22Evaluate 32=922Evaluate 42=1622

Finally, as there is no multiplication or division, complete addition and subtraction from left to right.

\begin{align*}9+16-9+6= & 22 \quad \text{Add }9+16=25\\ 25-9+6 = & 22 \quad \text{Subtract }25-9=16\\ 16+6 = & 22 \quad \text{Add }16+6=22\\ 22 = & 22 \quad \text{This is true}\\\end{align*}9+169+6=259+6=16+6=22=22Add 9+16=2522Subtract 259=1622Add 16+6=2222This is true

The answer is "The expression is true."

Example 4

Determine if the following is a true expression.

\begin{align*}6+3\times 2-5+\left(7-1\right)=19\end{align*}

First, evaluate the parentheses. 

\begin{align*}6+3\times 2-5+\left(7-1\right)= & 19 \quad \text{Evaluate }(7-1)=6\\ 6+3 \times 2 - 5 + 6 = & 19\\\end{align*} 

Next, there are no exponents to evaluate, so multiply. 

\begin{align*}6+3 \times 2-5+6= & 19 \quad \text{Multiply }3 \times 2=6\\ 6+6-5+6 = & 19\\\end{align*}

Then, complete the addition and subtraction in order from left to right.

\begin{align*}6+6-5+6= & 19 \quad \text{Add }6+6=12\\ 12-5+6 = & 19 \quad \text{Subtract }12-5=7\\ 7+6 = & 19 \quad \text{Add }7+6=13\\ \color{red}13 = & \color{red}19 \quad \color{red} \text{This is not a true statement}\\\end{align*}

The answer is "This expression is not true."

Review

Check each answer using order of operations. Write whether the answer is true or false.

  1. \begin{align*}4+5\times2+8-7=15\end{align*} 
  2. \begin{align*}4+3\times9+6-10=104\end{align*} 
  3. \begin{align*}6+2^2\times4+3\times6=150\end{align*} 
  4. \begin{align*}3+6\times3+9\times7-18=66\end{align*} 
  5. \begin{align*}7\times2^3+4-9\times3-8=25\end{align*} 
  6. \begin{align*}2\times3^3+7\times3=183\end{align*} 
  7. \begin{align*}4\times3+4^2-9+8=25\end{align*} 
  8. \begin{align*}3^2\times2^3+14-9=77\end{align*} 
  9. \begin{align*}6\times3^3-24+19\times2-4=310\end{align*} 
  10. \begin{align*}5\times2+5^2-10\times2-7 = 8\end{align*} 

Insert parentheses to make each expression true.

  1. \begin{align*}4+5-2+3-2=8\end{align*} 
  2. \begin{align*}2+3\times2-4=6\end{align*} 
  3. \begin{align*}1+9\times4\times3+2-1=110\end{align*} 
  4. \begin{align*}7+4\times3-5\times2=23\end{align*} 
  5. \begin{align*}2^2+5\times8-3+4=33\end{align*}  

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.12. 

Resources

 

 

 

 

 

 

Vocabulary

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.

Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.

Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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